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TRANSACTIONS OF THE SECTIONS. 



1. MATHEMATICS AND PHYSICS. 



MATHEMATICS. 



On the Application to Dynamics of a Getieral Mathematical 

 Method previously applied to Optics. By W. R. Hamilton, 

 M.R.I. A., Astronomer Royal for Ireland. 



The method is founded on a combination of the principles of 

 variations witli those of partial differentials, and may suggest 

 to analysts a separate branch of algebra, which may be called, 

 perhaps, the Calculus of Principal Functions ; because, in all 

 the chief applications of algebra to physics, and in a very ex- 

 tensive class of purely mathematical questions, it reduces the 

 determination of many mutually connected functions to the 

 search and study of one' principal or central relation. In ap- 

 plying this method to Dynamics, (having previously applied it 

 to Optics,) Professor Hamilton has discovered the existence of 

 a principal function, which, if its form were fully known, would 

 give, by its partial differential coefficients, all the intermediate 

 and all the final integrals of the known equations of motion. 



Professor Hamilton is of opinion that the mathematical ex- 

 planation of all the phaenomena of matter distinct from the 

 phaenomena of life, will ultimately be found to depend on the 

 pi'operties of systems of attracting and repelling points. And 

 he thinks that those who do not adopt this opinion in all its 

 extent, must yet admit the properties of such systems to be 

 more highly important in the present state of science, than 

 any other part of the application of mathematics to physics. 

 He therefore accounts it the capital problem of Dynamics, "to 

 determine the 3 n rectangular coordinates, or other marks of 

 position, of a free system of n attracting or repelling points, as 

 functions of the time," involving also 6 n initial constants, which 

 depend on the initial circumstances of the motion, and in- 



1834. 2 h 



